# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.

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### Constructing new complex manifolds out of old

It is not difficult to build new manifolds out of old in the smooth category, for example
taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth ...

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### Compact complex affine Kähler manifold is a torus

Before giving a motivation let me ask the precise question firstly.
By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which ...

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### Multiple mirrors phenomenon from SYZ and HMS perspective

There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...

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### Differences of $\omega$-plurisubharmonic functions

Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$.
A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...

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### Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...

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### Quotient of a smooth projective surface by an involution

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?

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### Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...

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### Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators.
For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...

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### What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.

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### Birational maps mapping ample class to ample class?

I refer to the paper "Normal Subgroups in the Cremona Group". In the last paragraph of the proof of proposition 5.13, the author wrote the following: "Assume now that $h\in \text{Bir}(X)$ preserves ...

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### Homogeneous Riemann Surfaces

A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification ...

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### Extending Normal Bundle of a Subvariety of $\mathbb P^n$

This question is related to, but not the same as, an earlier question: Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
Given a smooth projective variety $X\subset\mathbb P^n$, let $N_X$ be ...

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### Can we move curves which are members of very ample systems?

Let us take the second degree Hirzebruch surface F_2 which is a holomorphic CP^1 bundle over CP^1 having sections of self intersections +2 and -2. Let me denote the class of the -2 section by C and ...

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### Complex manifolds whose Hodge numbers are rigid under small deformations

Let $M$ be a closed complex manifold. Assume that for any family of closed complex manifolds over the unit disk containing $M$ as the central fiber, there exists a sufficiently small neighbourhood of ...

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### Holomorphic version of Darboux's theorem

I would like to ask if there is a holomorphic version of Darboux's theorem. More concretely, given a holomorphic symplectic manifold $(X, \omega)$ is there a local holomorphic symplectomorphism from $(...

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### Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...

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### Does there exist a curve which avoids a given countable union of small subsets?

Let $X$ be a projective variety over $\mathbb{C}$. Let $X_1, X_2, \ldots$ be proper closed subsets of $X$. Then $\cup_i X_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement.
...

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### Local behaviour of the moduli space of almost complex structures (up to conjugation)

Assume $M$ is a closed smooth manifold of real dimension $\geq 4$. What is known about the geometry of the "space" of almost complex structures up to conjugation by diffeomorphisms? There are quotes ...

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### Higher genus Gromov-Witten invariants and mirror symmetry

As a physicist, my understanding of mirror symmetry is very limited, and perhaps the most "mathematical" literature I have read on mirror symmetry is the book of M. Gross. In the genus-0 Gromov-Witten ...

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### Constancy of Hodge numbers in a family of compact complex manifolds

Does there exist a family of compact complex manifolds over unit disk such that the Hodge numbers are not constant in the family?
The answer is manifestly positive in complex dimension 1.
It is ...

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### Examples of manifolds that do not admit scalar flat metrics

The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:
(A) Every (smooth) function is a scalar curvature.
(B) The manifold is strongly ...

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### Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are ...

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### Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case

Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...

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### Reference on Complex Geometry

For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...

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### Properties of a particular Kummer Surface

Let $Y$ be the abelian variety $\mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i] $ where $\mathbb{Z}[i]$ denote the set of Gaussian integers in $\mathbb{C}$. Let $X$ be the quotient of $Y$ by ...

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### How much algebraic geometry do I need to study complex geometry?

As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...

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### Proper projections

Let $D \subset \mathbb{C}^k$ be your favorite complex domain.
Suppose we are given a proper holomorphic mapping $f \colon D \to \mathbb{C}^{k+2}$.
Let us take $k+1$ generic linear functions $l_i \...

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### Very ample linear systems - intersections with multiplicity >1

On a degree $n$ Hirzebruch surface $F_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F_n$. Let us take a member, $G$...

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### Is being of general type stable under generization

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
Definition. An integral projective ...

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### Compatible solution of PDE

Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...

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### Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.
$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...

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### Is it possible to glue together complex manifolds?

In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...

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### What are the automorphisms of a Grassmannian?

I want to know what are the holomorphic automorphisms of a Grassmannian. Can someone tell me this?

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### Equalizer of local analytic isomorphisms

Let $a,b : V\to W$ be two morphisms of smooth complex analytic spaces.
Assume $a$ and $b$ are local analytic isomorphisms.
Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic ...

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### A complex limit cycle not intersecting the real plane

Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow?
There is a regular leaf $L$ whose holonomy, along at least one closed curve ...

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### An almost complex structure on $S^2\times …\times S^2 / \mathbb{Z_2}$

Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...

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### Almost complex structure and intrinsic torsion

Given a $2m$-dimensional manifold $M$, an almost complex structure $J$ is equivalent to a $\text{GL}(m,\mathbb C)$-structure on $M$.
I wonder why the intrinsic torsion of the $\text{GL}(m,\mathbb C)$...

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### Compact Kaehler submanifolds of projectivized Hilbert space

If we take a separable complex Hilbert space $H$, its projective space $PH$ is an infinite-dimensional Kähler manifold in a fairly obvious sense (see below). Suppose $M \subset PH$ is a finite-...

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### Explicit descriptions of a flop

I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the ...

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### On $G$-gerbes over the punctured disk

Let $G$ be a finite (not necessarily abelian) group and let $\mathcal{X}\to D^*$ be a $G$-gerbe over the punctured disk $D^*$.
Is there a finite etale cover $D^*\to \mathcal{X}$?
I think of $G$-...

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### Does profinite completion commute with mapping spaces?

Does there exist a prime number $p$ and a smooth complex projective variety $X$ such that $F_{\infty p}\mathrm{Map}(B\mathbb{Z}/p\mathbb{Z}, X)$ is not weakly homotopy equivalent to $\mathrm{Map}(B\...

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### First Chern Class of Contact Structure which is not Torsion

Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...

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### (Singular) metric associated to the higher cohomology

Suppose $X$ is a smooth complex variety and $L$ is a line bundle with a metric $h_L$, then a section $s \in H^0(X, L)$ gives another metric $\tilde h_L:= e^{-\phi}h_L$ where $\phi=\log \|s\|^2_{h_L}$.
...

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### Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is ...

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### Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...

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### $c_2$ of Calabi-Yau three-folds

Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example?
...

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### h-principle for pairs

Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \...

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### Counting the number of poles for rational functions in a coordinate ring of a curve

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...

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### Complex manifolds as algebro-geometric objects

A result of Artin states that analytification of proper algebraic spaces over
$\mathbb{C}$ defines an an equivalence of the category of proper algebraic spaces with the category of Moishezon spaces. ...

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### Fuchsian groups of singly branched covers

Let $X/\mathbb{C}$ be an algebraic curve with genus $g \geq 2$. Then by the uniformization theorem, with $X(\mathbb{C})$ viewed as a Riemann surface, it can be realized as the quotient $\mathbb{H}/\...